A. Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these "maximally symmetric" spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic noncompact orbits. We also calculate the soldering form, the associated vielbein and any invariant aristotelian, galilean or carrollian structures. The (conformal) symmetries of the galilean and carrollian structures we determine are typically infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the space of invariant affine connections on each homogeneous spacetime and work out their torsion and curvature. 1 We refer to, e.g., [6] for further motivation and a (non-exhaustive) list of further references. While this work was under completion the interesting work [7] appeared which discusses similar aspects as this and our earlier work. Recently, also further interesting works, which fall in the realm of the kinematical Lie algebras and spacetimes, have appeared, see, e.g., [8,9,10,11,12,13,14,15,16,17]. 6 FIGUEROA-O'FARRILL, GRASSIE, AND PROHAZKA T . Simply-connected homogeneous (D + 1)-dimensional kinematical spacetimes Label D Non-zero Lie brackets in addition to [J, J] = J, [J, B] = B, [J, P] = P Comments S1 1 [H, B] = −P [B, B] = J [B, P] = H M S2 2 [H, B] = −P [H, P] = −B [B, B] = J [B, P] = H [P, P] = −J dS S3 1 [H, B] = −P [H, P] = B [B, B] = J [B, P] = H [P, P] = J AdS S4 1 [H, B] = P [B, B] = −J [B, P] = H E S5 1 [H, B] = P [H, P] = −B [B, B] = −J [B, P] = H [P, P] = −J S S6 1 [H, B] = P [H, P] = B [B, B] = −J [B, P] = H [P, P] = J

A. We classify N=1 d = 4 kinematical and aristotelian Lie superalgebras with spatial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quaternionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of 43 isomorphism classes of Lie superalgebras, some with parameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous (4|4)-dimensional superspaces, resulting in a list of 27 homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.

We study and classify Lie algebras, homogeneous spacetimes and coadjoint orbits ("particles") of Lie groups generated by spatial rotations, temporal and spatial translations and an additional scalar generator. As a first step we classify Lie algebras of this type in arbitrary dimension. Among them is the prototypical Lifshitz algebra, which motivates this work and the name "Lifshitz Lie algebras". We classify homogeneous spacetimes of Lifshitz Lie groups. Depending on the interpretation of the additional scalar generator, these spacetimes fall into three classes:(1) (d + 2)-dimensional Lifshitz spacetimes which have one additional holographic direction;(2) (d + 1)-dimensional Lifshitz-Weyl spacetimes which can be seen as the boundary geometry of the spacetimes in ( 1) and where the scalar generator is interpreted as an anisotropic dilation;(3) and (d + 1)-dimensional aristotelian spacetimes with one scalar charge, including exotic fractonlike symmetries that generalise multipole algebras. We also classify the possible central extensions of Lifshitz Lie algebras and we discuss the homogeneous symplectic manifolds of Lifshitz Lie groups in terms of coadjoint orbits. 3 6.3. a ± 4 and a ± 5 6.4. a 6 and a 7 7.FIGUEROA-O'FARRILL, GRASSIE, AND PROHAZKA